Ostrowski and Čebyšev type inequalities for interval-valued functions and applications

As an essential part of classical analysis, Ostrowski and Čebyšev type inequalities have recently attracted considerable attention. Due to its universality, the non-additive integral inequality takes several forms, including Sugeno integrals, Choquet integrals, and pseudo-integrals. Set-valued analysis, a well-known generalization of classical analysis, is frequently employed in studying mathematical economics, control theory, etc. Inspired by pioneering work on interval-valued inequalities, this paper establishes specific Ostrowski and Čebyšev type inequalities for interval-valued functions. Moreover, the error estimation to quadrature rules is presented as some applications for illustrating our results. In addition, illustrative examples are offered to demonstrate the applicability of the mathematical methods presented.


Introduction
Interval-valued functions hold immense mathematical and practical significance, occupying a central role across diverse academic disciplines.Their distinct properties and characteristics are integral to the study of interval optimization, interval differential equations, and random set analysis.In particular, interval-valued functions exhibiting integrability and differentiability play a crucial role in the abovementioned areas.Furthermore, these functions assume a prominent position within the fuzzy theory, as they enable the representation of fuzzy-valued functions through a collection of interval-valued functions, utilizing the notion of levels within a fuzzy interval.The main topic of this paper is Ostrowski and Čebys ˇev type inequalities.In 1882, Čebys ˇev [1] obtained an inequality: where f ; g : ½a; b� !R are absolutely continuous functions, and f 0 , g 0 2 L 1 ([a, b]).
Integral inequalities of the Ostrowski, Chebyshev, and Gruss type are well-known and appear in many areas of mathematics(for history and generalizations, see the renowned monograph [3], along with the papers [4][5][6][7]).Čebys ˇev and Ostrowski type inequalities, which have a very close relationship (see details in [8]), play an important role in many areas of mathematical applications and have received extensive attention from researchers.For example, Gru ¨ss [9] presented an inequality as follows: where f ; g : ½a; b� !R be two integrable functions with ϕ � f(x) � F, γ � g(x) � Γ for all x 2 [a, b] and �; F; g; G 2 R: Ujević [10] obtained the following Ostrowski type inequality: ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ðb À aÞkf 0 k where f : ½a; b� !R be a differentiable mapping with p is the best possible.For details of other Čebys ˇev and Ostrowski type inequalities, we refer the interested reader to [11][12][13][14][15][16][17][18][19][20].
Constructing a variety of integral inequalities is a contemporary concern.In recent years, a great amount of meaningful work has been completed using a variety of integrals, such as the Sugeno integral [21,22], the pseudo integral [23], and the Choquet integral [24], etc. Intervalvalued functions [25], as a concept of generalization of functions and an important mathematical topic, have become an increasingly vital instrument for solving practical issues, particularly in mathematical economics [26].Recent research has extended some traditional integral inequalities to the domain of interval-valued functions.Costa et al. [27] presented new interval versions of Minkowski and Beckenbach's integral inequalities.Hermite Hadamard, Jensen, and Ostrowski-type inequalities were proven in this generalization [28].Also, Hermite-Hadamard and Hermite-Hadamard-type inequalities were addressed using interval-valued Riemann-Liouville fractional integrals [29].Zhao et al. [30][31][32] studied Chebyshev type inequalities, Opial-type integral inequalities, and Jensen and Hermite-Hadamard type inequalities for interval-valued functions using the gH -differentiable or h-convex concept.Budaka et al. [33] derived novel fractional inequalities of the Ostrowski type for interval-valued functions, utilizing the definitions of gH-derivatives.Khan et al. [34] introduced log-h-convex fuzzy-interval-valued functions as a distinct class of convex fuzzy-interval-valued functions, employing a fuzzy order relation.This class served to establish Jensen and Hermite-Hadamard inequalities.
Incorporating the Ostrowski-type inequality within the realm of fuzzy-valued functions necessitated the utilization of the Hukuhara derivative, as demonstrated by Anastassiou [35].Fuzzy-valued functions, also known as functions with an interval value, were central to Anastassiou's [35] research.Fascinatingly, the fuzzy Ostrowski-type inequalities derived by Anastassiou [35] also extended their validity to interval-valued functions.For a comprehensive understanding of the constraints imposed by the concept of the H-derivative on interval-valued functions, it is worth examining the works of Bede and Gal [36] and Chalco-Cano et al. [37].Notably, recent contributions by Chalco-Cano et al. [38] have successfully established an Ostrowski type inequality catering specifically to generalized Hukuhara differentiable intervalvalued functions.The utmost significance of generalized Hukuhara differentiability as the most comprehensive concept for characterizing the differentiability of interval-valued functions has been underscored in prominent studies by Bede and Gal [36], Chalco-Cano et al. [39].
To the best of my knowledge, there is a gap in the literature regarding exploring Čebys ˇev and Ostrowski type inequalities utilizing generalized Hukuhara differentiation.Motivated by the existing literature, our primary objective is to introduce a novel approach that addresses the challenges associated with interval-valued functions.Specifically, we aim to present a series of interval-based variations of Čebys ˇev and Ostrowski type inequalities, utilizing the concept of generalized Hukuhara differentiability (as referenced in [39,[40][41][42]).This proposed approach offers several significant benefits: Enhanced Precision: By developing interval versions of Čebys ˇev and Ostrowski-type inequalities, our research enables a more precise analysis of interval-valued functions.This heightened precision can be precious in scientific and engineering fields where accuracy is paramount.
Improved Handling of Uncertainty: Interval-valued functions inherently capture uncertainty in data or model parameters.Using generalized Hukuhara differentiability, our methodology provides a robust framework for effectively managing this uncertainty.It mainly benefits decision-making processes, risk assessment, and optimization under uncertain conditions.
Broadened Applicability: The introduction of interval versions of these well-known inequalities expands the applicability of the existing theory to interval-valued functions.This extension not only enriches the field of interval analysis but also allows for new avenues of research and exploration of mathematical properties specific to interval-valued operations.
Estimating error in quadrature rules for gH-differentiable interval-valued functions is a fundamental application of these newly derived inequalities.The first application of Ostrowski's inequality to quadrature formulas was given in [43].(see also [44]).This paper also presented new quadrature procedures and error estimates as a theoretical application of this new fuzzy Čebys ˇev and Ostrowski type inequalities.When our goal is to obtain definite integrals of the form , classical quadrature rules, such as the trapezoidal and Simpson's rules, can be employed to approximate The research conducted by Chalco-Cano et al. [37] has highlighted that gH-differentiability in f does not imply the gH-differentiability of f and f .Consequently, the conventional error estimation methods that rely on differentiability cannot be directly utilized for these quadrature rules.In contrast, when working with interval-valued functions represented by f, leveraging established interval arithmetic techniques and software allows for approximating the definite integral R b a f ðtÞdt.For a comprehensive understanding of this approach, refer to the studies conducted by Moore [45] and Moore et al. [46].Given the widespread prevalence of interval-valued functions, acquiring the capability to estimate their errors and proficiently employ quadrature techniques becomes indispensable.
The study is organized as follows: Section 2 presents some relevant preliminaries.Section 3 presents a new Čebys ˇev type inequality and two Ostrowski type inequalities.Section 4 discusses the new estimation of quadrature rules which contain the mid-point quadrature rule as a special case based on the results in Section 3.

Preliminaries
Let R be the real line, and let R I denote the family of all closed interval of R, that is, Let A ¼ ½a; a� and B ¼ ½b; b�, the interval arithmetic are defined as follows: • Addition: • Multiplication: A � B ¼ AB ¼ ½minfab; ab; ab; abg; maxfab; ab; ab; abg�; • Generalized Hukuhara difference (gH-difference) [39,[40][41][42]: Now, we impose the Hausdorff metric or distance on R I , that is, Lemma 1 [47] H is a complete metric in R I and has the following properties (i) HðA; BÞ ¼ HðA� gH B; 0Þ; (ii) HðlA; lBÞ ¼ jljHðA; BÞ; l 2 R; Note that for any interval-valued function f : ½a; b� !R I , there exists two real functions f; f The functions f and f are called the lower and the upper functions of f, respectively.
We say an interval-valued function f is continuous at t 0 2 [a, b] if both f and f are continuous functions at t 0 2 [a, b].Denote by Cð½a; b�; R I Þ the space of all continuous interval-valued functions.Then, Cð½a; b�; R I Þ is a quasilinear space (see [47]) endowed with a quasi-norm k � k 1 given by kf k 1 ¼ sup t2½a;b� Hðf ðtÞ; 0Þ: ð6Þ Definition 2 [48] Let f : ½a; b� !R I be an interval-valued function.f is said to be Aumann integrable if the set S(f) of all integrable selectors of f, i.e., Lemma 3 [48] Let f ; g : ½a; b� !R I be two measurable and integrability bounded intervalvalued functions.Then for any Hðf ðtÞ; gðtÞÞdt.

Inequalities for interval-valued functions
This section presents Ostrowski and Čebys ˇev type inequalities for continuously gH-differentiable interval-value functions.First, we offer an Ostrowski type inequality.Theorem 7 Let f : ½a; b� !R I be an interval-value function.If its gH-derivative f 0 is continuous, then for any x 2 [a, b], Proof According to Lemmas 1 and 3, we have Furthermore, due to [49] [Theorem 4], one has On the other hand, Thus, it follows from ( 11)-( 13) that This ends the proof.

:
On the other hand, The graph of the two functions HðAðxÞ; BðxÞÞ and EðxÞ is given in Fig 1 .It is easy to see that HðAðxÞ; BðxÞÞ � EðxÞ; 8x 2 ½1; e�: So that the theorem 7 holds.In Theorem 7, if x ¼ aþb 2 , then we present another one generalization of inequality.Corollary 9 Let f : ½a; b� !R I be an interval-value function.If its gH-derivative f 0 is continuous, then Then, HðAðxÞ; BðxÞÞ ¼ j2 x ð2x À 1Þj: On the other hand, The graph of the two functions HðAðxÞ; BðxÞÞ and EðxÞ is given in Due to Lemma 11, On the other hand, from (2), it follows that Hence, This ends the proof.On the other hand, The graph of the two functions HðAðxÞ; BðxÞÞ and EðxÞ is given in Fig 3 .It is easy to see that HðAðxÞ; BðxÞÞ � EðxÞ; 8x 2 ½1; 2�: Now we are coming to the Čebys ˇev type inequality for interval-valued functions.Theorem 15 Let f : ½a; b� !R I be an interval-value function, g : ½a; b� !R a bounded function.If f has a continuous gH-derivative f 0 , then Proof By Lemmas 1 and 3, we have  Thus, it follows from ( 19) and ( 20) that According to Eq (21), the proof is therefore complete.

Remark 16
In Theorem 15, if g � 0 on [a, b], then the equality holds. Hence, Thanks to Theorem 15, we have another Ostrowski type inequality.Corollary 18 Let f : ½a; b� !R I be an interval-value function.If its second order gH-derivative f 00 is continuous, then for any x 2 [a, b], From Lemmas 1 and 3 and Theorem 15, it follows that According to (22), this ends the proof.Applications to the numerical quadrature rules where A R denotes the quadrature rule of Riemann-type defined by Proof From Lemmas 1 and Corollary 14, it follows that According to [49,Theorem 4] and Eq (24), we have The proof is therefore complete.Example 21 Considering f(x) = [x, x 2 + 2x], and t i ¼ i n ði ¼ 1; 2; :::; nÞ; h i ¼ 1 n : We can get kf 00 k 1 = 2, and So that the proposition 20 holds.
Example 22 Considering f(x) = [x 2 , 4x 2 ], and t i ¼ i n ði ¼ 1; 2; :::; nÞ; h i ¼ 1 n : We can get kf 00 k 1 = 8, and Hence, The graph of the two functions HðA R ; R b a f ðtÞdtÞ and y ¼ RðnÞ is given in Fig 6 .It is easy to see So that the proposition 20 holds.
Proposition 23 Let f : ½a; b� !R I be an interval-value function.If its second order gH-derivative f 00 is continuous, then for any K-partition P 2 PðI h ; xÞ, where S G denotes the generalized Riemann type quadrature rule defined by Proof According to Lemmas 1 and 3 and Corollary 18, we have The proof is therefore complete. The

Conclusion
This article presents a novel application of Ostrowski and Čebys ˇev type inequalities to gHdifferentiable interval-valued functions, expanding upon the existing theory.The findings of this study contribute significantly to the field of interval differential (or integral) inequalities and interval differential equations.By incorporating generalized Hukuhara differentiability and introducing interval-based variations of Ostrowski and Čebys ˇev type inequalities, our work advances the field of interval analysis, providing researchers and practitioners with powerful tools to address the complexities associated with interval-valued functions.Several related examples are solved to demonstrate the proposed method's effectiveness, showcasing its practical applicability.Additionally, this article introduces new error estimation techniques for fuzzy quadrature rules, showcasing a theoretical application of these mathematical techniques.In scenarios where the functions f and f lack differentiability, conventional error estimation methods for quadrature rules must be revised.Classical error estimation results, which rely on differentiability assumptions, are not applicable in these cases.However, by considering interval-valued functions and extending existing quadrature formulas for real functions, the approximation of R b a f ðtÞdt becomes feasible.Consequently, these extended quadrature formulas provide a valuable means of approximating the integral when dealing with interval-valued functions.
Furthermore, there are additional opportunities for further development and exploration.For instance, new versions of Ostrowski and Čebys ˇev type inequalities can be derived for various unique means, such as arithmetic mean, geometric mean, harmonic mean, and so on [17].In future research, it is intended to utilize these new and exciting inequalities for fuzzy-interval-valued functions.Additionally, it is anticipated that applying Ostrowski and Čebys ˇev type inequalities to set-valued and fuzzy set-valued functions will have implications in the fields of fluctuations, dynamic systems, hesitant fuzzy sets, and other related areas.

Fig 4 .
Fig 4. HðAðxÞ; BðxÞÞ (the red line), EðxÞ (the blue line) and Ê ðxÞ (the dash line) given in Example 19.https://doi.org/10.1371/journal.pone.0291349.g004 2n : The graph of the two functions HðA R ; R b a f ðtÞdtÞ and y ¼ RðnÞ is given in Fig 5.It is easy to see that H A R ;